We want to find the relationship between slope and a unit rate.

Unit Rate

A rate is a ratio that compares two quantities. Often we want to know how much of something is per $1$ unit of something else. A unit rate tells us the amount of one item compared to $1$ of another item. Let's look at some examples!

Scenario	Rate	Unit Rate
In a grocery store, $3$ kids bought $15$ candies.	$15$ candies per $3$ kids	$5$ candies per $1$ kid
Ana learns $20$ new words every $10$ minutes.	$20$ words per $10$ minutes $10$ words per $5$ minute	$2$ words per $1$ minute $120$ words per $1$ hour

Slope

Slope is the measure of the steepness of a line. Slope is the ratio of the

rise

(vertical change)

to the

run

(horizontal change) .

Where

rise

corresponds to the change in

y

and

run

corresponds to the change in

x .

m = \frac{rise}{run} = \frac{vertical change}{horizontal change}

As we can see, slope is a fraction. Using expansion or reduction, every fraction can be rewritten so that its denominator is

1 .

For example we can represent

2

as

\frac{2}{1},

- 1

as

\frac{- 1}{1},

and

\frac{5}{2}

as

\frac{5 / 2}{1}

or

\frac{2 . 5}{1} .

Comparison

Slope with denominator of $1$ is a rate that compares vertical change to $1$ unit of horizontal change. Therefore, in proportional relationships slope is the same as a unit rate. In relationships that are not proportional slope is not defined.

Exercise

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